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For how many integers \( x \) is the number \( x^4 - 51x^2 + 100 \) negative? | 10 | 0.166667 |
Given Sharon usually takes 4 hours to drive from her house to her workplace, and on one day she drives at her normal speed for half the distance but then reduces her speed by 30 miles per hour for the remaining half. If the total journey on that day took 330 minutes, calculate the total distance from Sharon's house to her workplace. | 280 | 0.916667 |
Dave, Emily, and Fiona take turns tossing an eight-sided die, starting with Dave, followed by Emily, and then Fiona. Determine the probability that Fiona is the first to toss an eight. | \frac{49}{169} | 0.75 |
The number $15!$ has a certain number of positive integer divisors. Determine the probability that one of its divisors is odd. | \frac{1}{12} | 0.333333 |
Find the sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers. | 50 | 0.916667 |
The product of two positive numbers is 16. The reciprocal of one of these numbers is 3 times the reciprocal of the other number. What is the sum of the two numbers? | \frac{16\sqrt{3}}{3} | 0.833333 |
In trapezoid $EFGH$, the sides $EF$ and $GH$ are equal, and the height from $F$ to $GH$ is 6 units. Given that the bottom base $GH$ is 12 units long, and the top base $EF$ is 10 units, find the perimeter of trapezoid $EFGH$. | 22 + 2\sqrt{37} \text{ units} | 0.666667 |
A sequence of $6$ numbers is formed by starting with two initial numbers. Each subsequent number in the sequence is the product of the two previous numbers. Given that the last two numbers of the sequence are $81$ and $6561$, find the value of the first number in the sequence. | \frac{1}{81} | 0.333333 |
Given that Theresa’s parents have promised to buy her tickets if she spends an average of 9 hours per week helping around the house for 6 weeks, and for the first 5 weeks she has helped 9, 12, 6, 13, and 11 hours, determine how many hours she must work in the final week to earn the tickets. | 3 | 0.666667 |
Let $M$ be the second smallest positive integer that is divisible by every positive integer less than 10 and includes at least one prime number greater than 10. Find the sum of the digits of $M$. | 18 | 0.833333 |
Given that nine unit squares are arranged in a 3x3 grid on the coordinate plane, with the lower left corner at the origin, and a line extending from (d,0) to (0,3) divides the entire configuration into two regions with equal areas, calculate the value of d. | 3 | 0.333333 |
The number $15!$ has several positive integer divisors. Find the probability that a randomly chosen divisor is odd. | \frac{1}{12} | 0.083333 |
Given that Liam bought some pencils costing more than a penny each at the school bookstore and paid $\textdollar 2.10$, and Mia bought some of the same pencils and paid $\textdollar 2.82$, determine the number of pencils that Mia bought compared to Liam. | 12 | 0.75 |
Mr. Green measures his rectangular garden by taking 18 steps by 25 steps. Each of Mr. Green's steps is 2.5 feet long. He expects three-quarters of a pound of potatoes per square foot from his garden. Calculate the area of the garden in square feet, and then find the total weight of potatoes that Mr. Green expects from his garden. | 2109.375 | 0.833333 |
Liam has two older twin sisters. The product of their three ages is 144. Find the sum of their three ages. | 16 | 0.583333 |
Given the sequence of squares with each side length one tile more than the previous square, calculate how many more tiles the eighth square requires than the seventh square. | 15 | 0.916667 |
Given that bricklayer Brenda would take 8 hours to build a wall alone, and bricklayer Brandon would take 12 hours to build it alone, and working together, they build the wall in 6 hours with a 15 bricks per hour reduction in their combined output, find the total number of bricks in the wall. | 360 | 0.583333 |
Given that Mrs. Blue's rectangular garden measures 10 steps by 30 steps, with each step being 3 feet long, determine the total weight of potatoes she expects from her garden, knowing that only 90% of the garden space is suitable for planting and she expects three-quarters of a pound of potatoes per square foot. | 1822.5 | 0.75 |
How many real numbers \(x\) satisfy the equation \(3^{2x+2} - 3^{x+4} + 5 \cdot 3^x - 15 = 0\)? | 1 | 0.916667 |
The basic subscription price of the online streaming service is $15 per month. Calculate the maximum percentage decrease in the number of subscribers that the service can tolerate in order to keep their total income at least the same despite a 20% price increase. | 16.67\% | 0.916667 |
Determine the minimum value of $\frac{3x^2 + 6x + 19}{8(1+x)}$ for $x \ge 0$. | \sqrt{3} | 0.583333 |
Given that $\frac{2}{3}$ of the marbles are blue and the rest are red, calculate the fraction of marbles that will be red if the number of red marbles is doubled and the number of blue marbles remains the same. | \frac{1}{2} | 0.833333 |
Let $x$ be the smallest real number greater than 0 such that $\cos(x) = \cos(x^2)$, where the arguments are in radians. Find the value of $x$. | 1 | 0.75 |
Calculate $\frac{(0.4)^4}{(0.04)^3}$. | 400 | 0.916667 |
A car travels the first part of its journey, covering a distance of $b$ miles at a speed of 50 mph, and then completes another part where it travels for $2n+b$ miles at a speed of 75 mph. Calculate the total time taken by the car to complete its journey. | \frac{5b + 4n}{150} | 0.083333 |
Given that Mr. A initially owns a home worth $\$15,000$, he sells it to Mr. B at a $20\%$ profit, then Mr. B sells it back to Mr. A at a $15\%$ loss, then Mr. A sells it again to Mr. B at a $10\%$ profit, and finally Mr. B sells it back to Mr. A at a $5\%$ loss, calculate the net effect of these transactions on Mr. A. | 3541.50 | 0.083333 |
Homer started peeling a pile of 50 potatoes at the rate of 4 potatoes per minute. Five minutes later, Christen joined him and peeled at the rate of 6 potatoes per minute. Determine the total number of potatoes peeled by Christen. | 18 | 0.916667 |
What is the value of $\frac{(2222 - 2002)^2}{144}$? | \frac{3025}{9} | 0.083333 |
A top hat contains 5 red chips and 4 green chips. Chips are drawn randomly, one at a time without replacement, until all 5 of the reds are drawn or until all 4 green chips are drawn. Calculate the probability that all 5 red chips are drawn. | \frac{4}{9} | 0.083333 |
At 3:30 o'clock, find the angle between the hour and minute hands of a clock. | 75^{\circ} | 0.916667 |
Eight students, including Abby and Bridget, are to be seated randomly in two rows of four for a group photo as shown:
\begin{eqnarray*}
\text{X}&\quad\text{X}\quad&\text{X}\quad&\text{X} \\
\text{X}&\quad\text{X}\quad&\text{X}\quad&\text{X}
\end{eqnarray*}
What is the probability that Abby and Bridget are seated adjacent to each other either in the same row or the same column?
A) $\frac{1}{7}$
B) $\frac{2}{9}$
C) $\frac{3}{14}$
D) $\frac{4}{15}$
E) $\frac{5}{14}$ | \frac{5}{14} | 0.166667 |
Given the numbers $2^9+1$ and $2^{17}+1$, inclusive, calculate the number of perfect cubes between them. | 42 | 0.75 |
Given the total degree measure of the interior angles of a convex polygon is $2083^\circ$, determine the degree measure of the omitted angle. | 77^\circ | 0.833333 |
Given the equation \(\frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}} = 1001\), find the value of \(\frac{x+y}{x-y}\). | -\frac{1}{1001} | 0.916667 |
Given that $a @ b = \frac{a \times b}{a + b}$ for $a,b$ positive integers, and further operation defined as $c # d = c + d$, calculate the value of $3 @ 7 \# 4$. | \frac{61}{10} | 0.25 |
The population of a village is $800$, and the graph indicates that the number of females is divided into four equal parts, with three of these parts representing the females. Determine the number of males in the village. | 200 | 0.833333 |
A piece of wood of uniform density in the shape of a right triangle with base length $3$ inches and hypotenuse $5$ inches weighs $12$ ounces. Another piece of the same type of wood, with the same thickness, also in the shape of a right triangle, has a base length of $5$ inches and a hypotenuse of $7$ inches. Calculate the approximate weight of the second piece. | 24.5 | 0.583333 |
A point is chosen at random from within a circular region with radius $3$, calculate the probability that the point is closer to the center of the region than it is to the boundary of the region. | \frac{1}{4} | 0.916667 |
Given that Mr. Blue receives a $15\%$ raise every year, calculate the percentage increase in his salary after five such raises. | 101.14\% | 0.916667 |
In a classroom with \( n > 15 \) students, the average score of a quiz is 10. A subset of 15 students have an average score of 17. Determine the average of the scores of the remaining students in terms of \( n \). | \frac{10n - 255}{n - 15} | 0.833333 |
In a $60$-question multiple choice contest, students receive $5$ points for a correct answer, $0$ points for an answer left blank, and $-2$ points for an incorrect answer. If Carla's total score on the contest was $150$, find the maximum number of questions that Carla could have answered correctly. | 38 | 0.916667 |
Given points P(0, -3) and Q(5, 3) in the xy-plane; point R(x, m) is taken so that PR + RQ is a minimum where x is fixed to 3, determine the value of m. | \frac{3}{5} | 0.166667 |
(3+13+23+33+43)+(11+21+31+41+51) = | 270 | 0.833333 |
What is the largest quotient that can be formed using two numbers chosen from the set $\{ -30, -5, -3, 1, 3, 10, 15 \}$? | 15 | 0.666667 |
Find the sum of all real numbers $x$ for which the median of the numbers $3, 7, 9, 20,$ and $x$ is equal to the mean of those five numbers and $x$ is less than $9$. | -4 | 0.5 |
Find the fraction exactly halfway between $\frac{1}{4}$, $\frac{1}{6}$, and $\frac{1}{3}$. | \frac{1}{4} | 0.833333 |
What is the tens digit of $(25! - 20!)$? | 0 | 0.916667 |
Given a grid arrangement of eight regular squares surrounding a central square of side length 2, find the area of triangle DEF, where D, E, and F are the centers of three adjacent outer squares. | 2 | 0.5 |
How many integer values of $x$ satisfy $|x| < 4\pi + 1$? | 27 | 0.916667 |
Given that there are 21 students in Dr. Smith's physics class, the average score before including Simon's project score was 86. After including Simon's project score, the average for the class rose to 88. Calculate Simon's score on the project. | 128 | 0.083333 |
The line joining the midpoints of the diagonals of a trapezoid has length $4$. If the longer base is $100$, determine the length of the shorter base. | 92 | 0.666667 |
Evaluate $\frac{900^2}{153^2-147^2}$. | 450 | 0.916667 |
A positive integer $n$ has $72$ divisors and $5n$ has $90$ divisors. Determine the greatest integer $k$ such that $5^k$ divides $n$. | 3 | 0.75 |
Given that the town experiences two consecutive years of population increase by 20%, followed by two consecutive years of population decrease by 30%, calculate the net percentage change in the population over these four years, rounded to the nearest percent. | -29\% | 0.833333 |
Calculate the value of $(2^3 - 3 + 5^3 - 0)^{-1} \times 7$. | \frac{7}{130} | 0.916667 |
An integer $N$ is selected at random from the range $1 \leq N \leq 1950$. Calculate the probability that the remainder when $N^{14}$ is divided by $5$ is $1$. | \frac{2}{5} | 0.833333 |
Calculate the difference $(2001 + 2002 + 2003 + \cdots + 2100) - (51 + 53 + 55 + \cdots + 149)$. | 200050 | 0.916667 |
Given a circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 40^{\circ}$, and $\angle DOB = 30^{\circ}$. Find the ratio of the area of the smaller sector $COD$ to the area of the circle. | \frac{11}{36} | 0.25 |
A circle of radius 7 is inscribed in a rectangle. The ratio of the length to the width of the rectangle is 3:1. Calculate the area of the rectangle. | 588 | 0.833333 |
Given Tom is currently $T$ years old, which is also equal to the sum of the ages of his four children. Three years ago, Tom's age was three times the sum of his children's ages. Determine the value of $\frac{T}{3}$. | 5.5 | 0.666667 |
Calculate the value of $3^{\left(1^{\left(0^8\right)}\right)}+\left(\left(3^1\right)^0\right)^8$. | 4 | 0.916667 |
Given the first three terms of a geometric progression are $2^2$, $2^{\frac{3}{2}}$, and $2$, respectively, find the fourth term. | \sqrt{2} | 0.75 |
Gary the gazelle takes 55 equal jumps to navigate between consecutive street lamps on a city park path, while Zeke the zebra covers the same distance in 15 equal strides. The distance to the 26th lamp from the start is 2640 feet. Calculate how much longer Zeke's stride is than Gary's jump. | 5.12 | 0.166667 |
Find the greatest number of consecutive non-negative integers whose sum is $120$. | 16 | 0.916667 |
Given the equation $\sin(3x) = \cos(x)$ on the interval $[0, 2\pi]$, find the number of solutions. | 6 | 0.666667 |
Given the range of $10000$ to $99999$, inclusive, determine the number of $5$-digit positive integers having only even digits that are divisible by $5$. | 500 | 0.916667 |
A $4 \times 4$ square is partitioned into $16$ unit squares. Each unit square is painted either white or black with each color being equally likely, independently and at random. The square is then rotated $180\,^{\circ}$ about its center, and every white square that lands in a position formerly occupied by a black square is painted black, while black squares moved to a position formerly occupied by white squares are turned white. Determine the probability that the entire grid is now black. | \frac{1}{65536} | 0.083333 |
Let $x$ be a real number selected uniformly at random between 0 and 1000. If $\lfloor \sqrt{x} \rfloor = 14$, find the probability that $\lfloor \sqrt{10x} \rfloor = 44$. | \frac{13}{58} | 0.75 |
Given that a four-digit integer is between 1000 and 9999, has only even digits, is divisible by both 5 and 10, and the hundreds digit is 0, determine the number of such integers. | 20 | 0.583333 |
Four years ago, Liam was twice as old as his sister Mia. Six years before that, Liam was three times as old as Mia. Determine the number of years until the ratio of their ages will be 3:2. | 8 | 0.25 |
Paul originally had enough paint for 50 identically sized rooms, but lost 5 cans on the way. Determine the number of cans of paint he used for the remaining 40 rooms. | 20 | 0.75 |
A pentagon is inscribed in a circle. Find the sum of the angles inscribed in the five arcs cut off by the sides of the pentagon. | 180^\circ | 0.916667 |
How many integer values of $x$ satisfy $|x|<4\pi$? | 25 | 0.916667 |
The numbers $\log(a^2b^4)$, $\log(a^6b^9)$, and $\log(a^{10}b^{14})$ form the first three terms of an arithmetic sequence, and the $10^\text{th}$ term of this sequence is $\log(a^n)$. Determine the value of $n$. | 38 | 0.75 |
Given that Jackie has $40$ thin rods, one of each integer length from $1 \text{ cm}$ through $40 \text{ cm}$, with rods of lengths $5 \text{ cm}$, $12 \text{ cm}$, and $20 \text{ cm}$ already placed on a table, find the number of the remaining rods that she can choose as the fourth rod to form a quadrilateral with positive area. | 30 | 0.083333 |
Given two numbers chosen from the set $\{-25, -4, -1, 1, 3, 9\}$, determine the largest quotient that can be formed. | 25 | 0.833333 |
Given the equation $16^{-3} = \frac{4^{60/y}}{4^{32/y} \cdot 16^{24/y}}$, find the value of y that satisfies this equation. | \frac{10}{3} | 0.75 |
The smallest positive integer that is neither prime nor square and that has no prime factor less than 70. | 5183 | 0.75 |
Given that in a class test, $15\%$ of the students scored $60$ points, $50\%$ scored $75$ points, $20\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test. | 2.75 | 0.583333 |
The sum of three numbers is $125$. The ratio of the first to the second is $\frac{3}{4}$, and the ratio of the second to the third is $\frac{7}{6}$. Find the second number. | \frac{3500}{73} | 0.5 |
Calculate the result of $\frac{3}{10} + \frac{5}{100} - \frac{2}{1000}$. | 0.348 | 0.333333 |
Given Alex, Jamie, and Casey play a game over 8 rounds, and for each round, the probability Alex wins is $\frac{1}{3}$, and Jamie is three times as likely to win as Casey, calculate the probability that Alex wins four rounds, Jamie wins three rounds, and Casey wins one round. | \frac{35}{486} | 0.75 |
Determine the value of $x$ that gives the minimum value of the quadratic function $x^2 + px + qx$. | -\frac{p+q}{2} | 0.583333 |
In the equation $\frac{x(x - 2) - (m + 2)}{(x - 2)(m - 2)} = \frac{x}{m}$, find the value of $m$ for which the roots are equal.
**A)** $m = -\frac{1}{2}$
**B)** $m = 1$
**C)** $m = 0$
**D)** $m = -1$
**E)** $m = -\frac{3}{2}$ | \textbf{(E)}\ m = -\frac{3}{2} | 0.083333 |
Given that the plane is tiled by alternative patterns of congruent squares and congruent pentagons, where each large square is divided into 16 smaller squares, out of which 5 smaller squares are turned into congruent pentagons, calculate the percent of the plane that is enclosed by the pentagons. | 31.25\% | 0.916667 |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$? | 10 | 0.75 |
The average age of the members of the family is 25, the mother is 45 years old, and the average age of the father and children is 20. If $x$ is the number of children in the family, determine the value of $x$. | 3 | 0.666667 |
Given a cyclic quadrilateral $ABCD$ with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=80^\circ$ and $\measuredangle ADC=110^\circ$, find the measure of $\measuredangle EBC$. | 110^\circ | 0.083333 |
Given that a tiled surface consists of $n^2$ identical blue square tiles, each measuring $s$ inches per side, and each surrounded by a yellow border $d$ inches wide, forming an overall large square pattern, and $n=30$ being observed to cover $81\%$ of the area of the entire square region, determine the ratio $\frac{d}{s}$. | \frac{1}{18} | 0.083333 |
Given that a circle is divided into 15 sectors and the central angles of these sectors form an arithmetic sequence, find the degree measure of the smallest possible sector angle. | 3 | 0.583333 |
How many whole numbers between 200 and 500 contain the digit 3? | 138 | 0.083333 |
A 3x3x3 cube is assembled from $27$ standard dice. Each die has the property that opposite faces sum to $7$. Calculate the largest possible sum of all of the values visible on the $6$ faces of the cube. | 288 | 0.083333 |
Alice has two pennies, three nickels, four dimes, and two quarters in her purse. Calculate the percentage of one dollar that is in her purse. | 107\% | 0.75 |
The real number \(x\) satisfies the equation \(x + \frac{1}{x} = \sqrt{3}\). Evaluate the expression \(x^{7} - 5x^{5} + x^{2}\). | -1 | 0.166667 |
A digital watch now displays time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits when it displays time in this format, where the hour ranges from 00 to 23 and the minutes range from 00 to 59. | 24 | 0.083333 |
Given a deck with five red cards labeled $1, 2, 3, 4, 5$ and four green cards labeled $1, 2, 3, 4$, find the probability of drawing a winning pair, where a winning pair is defined as two cards of the same color or two cards with the same number. | \frac{5}{9} | 0.75 |
Given that $\angle XYZ = 36^\circ$ and $\angle XYW = 15^\circ$, calculate the smallest possible degree measure for $\angle WYZ$. | 21^\circ | 0.916667 |
Evaluate the expression $(3(3(3(3(3 - 2 \cdot 1) - 2 \cdot 1) - 2 \cdot 1) - 2 \cdot 1) - 2 \cdot 1)$. | 1 | 0.666667 |
Calculate the sum of the first one hundred positive odd integers subtracted from the sum of the first one hundred positive even integers. | 100 | 0.916667 |
Given that ten friends dined at a restaurant and Sara's nine friends each paid an extra $3 to cover her part of the bill, determine the total value of the bill. | 270 | 0.666667 |
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